Method for operating a hydrocarbon deposit by injection of a gas in foam form

ABSTRACT

Method for operating a hydrocarbon deposit by injection of gas in foam form, comprising a step of determination of a model of displacement of the foam, this model being a function of an optimal mobility reduction factor of the gas and of at least one interpolation function dependent on a parameter and constants to be calibrated. 
     The mobility reduction factor of the gas is determined and the constants of at least one interpolation function are calibrated from experimental measurements comprising injections of gas in non-foaming form and in foam form into a sample of the deposit for different values of the parameter relative to the function considered, and measurements of headloss corresponding to each value of the parameter of the interpolation function considered. The calibration of the constants is performed interpolation function by interpolation function. 
     Applicable notably to oil exploration and operation.

The present invention relates to the field of the operation of a fluidcontained in an underground formation, more particularly the assistedrecovery of a fluid, such as a hydrocarbon fluid, by foam injection.

The operation of an oil reservoir by primary recovery consists inextracting, via a so-called production well, the oil present in thereservoir through the effect of overpressure prevailing naturally in thereservoir. This primary recovery makes it possible to access only asmall quantity of the oil contained in the reservoir, around 10 a 15% atthe very most.

To make it possible to continue to extract the oil, secondary productionmethods are employed, when the pressure of the reservoir becomesinsufficient to displace the oil still in place. In particular, a fluidis injected (re-injection of the water produced, diluted or not,injection of sea or river water, or even injection of gas, for example)into the hydrocarbon reservoir, in order to exert, in the reservoir, anoverpressure specifically to drive the oil to the production well orwells. One standard technique in this context is water injection (alsoreferred to as “waterflooding”), in which large volumes of water areinjected under pressure into the reservoir via injection wells. Theinjected water drives a part of the oil that it encounters and pushes itto one or more production producing wells. The secondary productionmethods such as water injection however, make it possible to extractonly a relatively small part of the hydrocarbons in place (typicallyaround 30%). This partial scavenging is due in particular to thetrapping of the oil by capillary forces, to the differences in viscosityand in density that exist between the injected fluid and thehydrocarbons in place, and to heterogeneity on microscopic ormacroscopic scales (scale of the pores and also scale of the reservoir).

To try to recover the rest of the oil, which remains in the undergroundformations after the implementation of the primary and secondaryproduction methods, there are various so-called assisted recoverytechniques (known by the acronym “EOR”, which stands for “Enhanced OilRecovery”). Among these techniques, techniques can be cited that aresimilar to the abovementioned water injection, but that employ a waterincluding additives such as, for example, soluble surface active agentsin the water (this is called “surfactant flooding”). The use of suchsurface active agents induces in particular a reduction of the water/oilinterfacial tension, which is specifically to ensure a more effectivedriving of the oil trapped in constrictions of pores.

Also known is assisted recovery by injection of gases, miscible or not(natural gas, nitrogen or CO₂). This technique makes it possible tomaintain the pressure in the oil reservoir during its operation, but canalso make it possible, in the case of miscible gases, to mobilize thehydrocarbons in place and thus improve the flow rate. A gas commonlyused is carbon dioxide when it is available at low cost.

Also known are alternative techniques relying on an injection of foaminto the oil reservoir. Because of its high apparent viscosity, the foamis considered as an alternative to gas as injection fluid in hydrocarbonreservoirs. The mobility of the foam is thus reduced relative to gaswhich, for its part, tends to segregate and perforate rapidly in theproducing wells, notably in the heterogeneous and/or thick reservoirs.Assisted recovery by foam injection is particularly attractive becauseit requires the injection of smaller volumes than for other assistedrecovery methods based on non-foaming fluids.

STATE OF THE ART

The following documents will be cited hereinbelow in the description:

-   Ma, K., Lopez-Salinas, J. L., Puerto, M. C., Miller, C. A.,    Biswal, S. L., Hirasaki, G. J., 2013. Estimation of Parameters for    the Simulation of Foam Flow through Porous Media. Part 1: The    Dry-Out Effect. Energy & Fuels 27, 2363-2375 (ACS Publications).-   Farajzadeh, R., Lotfollahi, M., Eftekhari, A. A., Rossen, W. R. and    Hirasaki, G. J., 2015. Effect of Permeability on Implicit-Texture    Foam Model Parameters and the Limiting Capillary Pressure. Energy    Fuels 29, 3011-3018 (ACS Publications).-   Kapetas, L., Vincent-Bonnieu, S., Farajzadeh, R., Eftekhari, A. A.,    Mohd-Shafian, S. R., Kamarul Bahrim, R. Z. and Rossen, W. R., 2015.    Effect of Permeability on Foam-Model Parameters—An Integrated    Approach from Coreflood Experiments through to Foam Diversion    Calculations. 18th European Symposium on IOR, Dresden, 14-16 April.

Oil operation of a deposit consists in selecting the areas of thedeposit that exhibit the best oil potential, in defining optimaloperating schemes for these areas (notably using a numerical simulationof the flows in the deposit, in order to define the type of recovery,the number and positions of the operation wells, allowing for an optimalhydrocarbon recovery), in drilling operation wells and, in general, inputting in place the production infrastructures necessary to thedevelopment of the deposit.

Defining an operating scheme of an oil reservoir including a step ofassisted recovery by foam injection may entail numerically simulating,in the most realistic way possible, the flows in the presence of foam inthe reservoir considered. Such a simulation is performed using a flowsimulator comprising a model of displacement of the foam.

Such a model may involve evaluating the performance levels of the foamin terms of mobility reduction. In general, this estimation involvesconducting laboratory trials consisting in measuring the headlosses indisplacements of foam on the one hand, of water and of non-foaming gason the other hand in a sample of the oil reservoir. Then, this model ofdisplacement of the foam, representative of the flows on a laboratoryscale, is calibrated to the scale of the reservoir before carrying outthe numerical simulations of the flows, in order to predict the benefitaccrued by the injection of the foam in terms of improvement of theeffectiveness of displacement of the fluids in place.

The foam displacement models used by the industry are relatively simplemodels which, subject to the conditions of existence of the foam,simulate only the effects of the foam in terms of mobility reduction andnot the foam generation-destruction processes. Generally, the foamdisplacement models depend nonlinearly on numerous parameters(calibration constants). Determining the parameters of this modeltherefore involves solving a nonlinear inverse problem. However, thecomplexity of the displacement of a foam in a confined environment thatany natural porous medium constitutes makes the modeling difficultbecause the great number of parameters influencing the foam can lead toindeterminacies (multiple solutions).

The approach proposed in the document (Ma et al., 2013) is known, whichconsists in simultaneously determining certain parameters of the foamdisplacement model by a graphic approach, completed by a numericaladjustment.

Also known is the technique proposed in the document (Farajzadeh et al.,2015) which proceeds with the determination of the unknown parameters(calibration constants) of the foam displacement model by an iterativeleast squares approach. However, since the problem posed is nonlinearwith respect to these unknowns, there is no unique solution, or in otherwords, the parameters thus determined are one solution out of otherpossible ones (see for example Kapetas et al., 2015).

The method according to the invention aims to determine, pragmatically,the parameters of the foam displacement model. Unlike the existingmethods, the method according to the invention consists, fromexperimental data, in a sequential adjustment of the parameters of thefoam model, and not in an overall adjustment. Thus, the method accordingto the invention makes it possible to minimize the numeric adjustments,by trying to extract the maximum of information on the dynamic behaviorof foam from experimental data.

THE METHOD ACCORDING TO THE INVENTION

Thus, the present invention relates to a method for operating anunderground formation comprising hydrocarbons, by means of an injectionof an aqueous solution comprising a gas in foam form and a flowsimulator relying on a displacement model of said gas in foam form, saiddisplacement model being a function of an optimal mobility reductionfactor of said gas and of at least one interpolation function of saidoptimal mobility reduction factor, said interpolation function being afunction of at least one parameter relating to at least onecharacteristic of the foam and of at least one constant. The methodaccording to the invention is implemented from at least one sample ofsaid formation, measurements of conventional relative permeabilities tosaid gas in non-foaming form and measurements of conventional relativepermeabilities to said aqueous phase, and comprises at least thefollowing steps:

-   -   A. said displacement model of said simulator is determined        according to at least the following steps:        -   i. a plurality of values of said parameter is defined            relative to at least one of said interpolation functions, an            injection is performed into said sample of said gas in            non-foaming form and of said gas in foam form according to            said values of said parameter relative to said function, and            a headloss with foam and a headloss without foam are            measured respectively for each of said values of said            parameter relative to said function;        -   ii. from said measurements of headloss relative to said            interpolation function, an optimal value of said parameter            relative to said function is determined, said optimal value            making it possible to maximize a ratio between said            headlosses without foam and said headlosses with foam            measured for said function;        -   iii. from said measurements of headloss performed at said            optimal value determined for at least said interpolation            function, from said measurements of conventional relative            permeabilities to said gas in non-foaming form and from said            measurements of conventional relative permeabilities to said            aqueous phase, said optimal mobility reduction factor is            determined;        -   iv. for at least said interpolation function, from said            optimal mobility reduction factor, from said measurements of            headloss relative to said interpolation function, from said            measurements of conventional relative permeabilities to said            gas in non-foaming form and from said measurements of            conventional relative permeabilities to said aqueous phase,            said constants of said interpolation function are            calibrated;    -   B—from said displacement model and from said flow simulator, an        optimal operating scheme for said deposit is determined and said        hydrocarbons are exploited.

According to one implementation of the invention, said displacementmodel of the foam can be expressed in the form:

k _(rg) ^(FO)(S _(g))=FM k _(rg)(S _(g)),

in which k_(rg) ^(FO)(S_(g)) is the relative permeability to said gas infoam form for a given gas saturation value S_(g), k_(rg)(S_(g)) is therelative permeability to said non-foaming gas for said gas saturationvalue S_(g), and FM is a functional expressed in the form:

${F\; M} = \frac{1}{\left( {M^{opt} - 1} \right)*{\prod\limits_{k}\; F_{k}}}$

in which M^(opt) is said optimal mobility reduction factor of said gasand F_(k) is one of said interpolation functions, with k≧1.

According to one embodiment of the invention, there can be four of saidinterpolation functions and said parameters of said functions can be afoaming agent concentration, a water saturation, an oil saturation, anda gas flow rate.

Advantageously, said interpolation function F_(k) of a parameter V_(k)can be written in the form:

${F_{k}\left( V_{k} \right)} = \frac{M_{k}^{opt} - 1}{M^{opt} - 1}$

in which M^(opt) is said optimal mobility reduction factor and M_(k)^(opt) is an optimal mobility reduction factor for said parameter V_(k).

According to one implementation of the invention, prior to the stepiii), optimal conditions can be defined as corresponding to said optimalvalues determined for each of said interpolation functions, said gas innon-foaming form and said gas in foam form can be injected into saidsample according to said optimal conditions, and a headloss with foamand a headloss without foam can be respectively measured.

Advantageously, said constants of at least one of said interpolationfunctions can be calibrated by a least squares method, such as aninverse method based on the iterative minimization of a functional.

Other features and advantages of the method according to the inventionwill become apparent on reading the following description of nonlimitingexemplary embodiments, with reference to the figure attached anddescribed hereinbelow.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 presents an example of the trend of a mobility reduction factor Ras a function of the gas flow rate Q.

DETAILED DESCRIPTION OF THE METHOD

In general, one of the objects of the invention relates to a method foroperating an underground formation comprising hydrocarbons, by means ofan injection of an aqueous solution comprising a gas in foam form, andin particular the determination of an optimal operating scheme for theunderground formation studied. In particular, the method according tothe invention targets the determination of the parameters of a model ofdisplacement of the gas in foam form. Hereinbelow, foam describes aphase dispersed in another phase by the addition of a foaming agent inone of the two phases. One of the phases can be water and the otherphase is a gas, such as natural gas, nitrogen or CO_(2.)

The method according to the invention requires the availability of:

-   -   a sample of the underground formation studied, taken by in situ        coring for example;    -   a flow simulator relying on a model of displacement of the gas        in foam form (see below);    -   measurements of conventional relative permeabilities to the gas        in non-foaming form and measurements of conventional relative        permeabilities to the aqueous phase: these can be measurements        performed expressly for the requirements of the method according        to the invention (the expert has perfect knowledge of how to        conduct such laboratory experiments), but they can also be        analytical functions calibrated from correlations well known to        the expert.

Thus, the method according to the invention requires the availability ofa flow simulator comprising a model of displacement of the foam.According to the invention, the model of displacement of the foam relieson the hypothesis that the gas present in foam form has its mobilityreduced by a given factor in set conditions of formation and of flow ofthe foam. The formulation of such a model, used by many flow simulators,consists in modifying the relative permeabilities to the gas when thegas is present in foam form which, for a given gas saturation S_(g), isexpressed according to a formula of the type:

k _(rg) ^(FO)(S _(g))=FM k _(rg)(S _(g))  (1)

in which k_(rg) ^(FO)(S_(g)) is the relative permeability to the gas infoam form, that is expressed as the product of a function FM by therelative permeability to the non-foaming gas k_(rg)(S_(g)) for the samegas saturation value S_(g) (later denoted S_(rg) ^(FO)). One assumptionunderpinning the current foam models is that the relative permeabilityto water (or to liquid by extension) is assumed unchanged, whether thegas is present in continuous phase form or in foam form. Given thisassumption, the gas mobility reduction functional, hereinafter denotedFM, is expressed according to a formula of the type:

$\begin{matrix}{{F\; M} = \frac{1}{1 + {\left( {M_{mod}^{opt} - 1} \right)*{\prod\limits_{k}\; {F_{k}\left( V_{k} \right)}}}}} & (2)\end{matrix}$

in which:

-   -   M_(mod) ^(opt) is the optimal mobility reduction factor, that is        to say the ratio of the relative permeabilities to the gas        (k_(rg)) and to the foam (k_(rg) ^(FO)) in optimal conditions        for reducing the mobility of the gas, that is to say the        conditions in which the terms F_(k)(V_(k)) defined hereinbelow        have the value 1, i.e.:

$\begin{matrix}{M_{mod}^{opt} = {\frac{k_{rg}\left( S_{g,{opt}}^{FO} \right)}{k_{rg}^{FO}\left( S_{g,{opt}}^{FO} \right)} = \frac{1}{F\; M_{opt}}}} & (3)\end{matrix}$

-   -   the terms F_(k)(V_(k)) (with k equal to or greater than 1) are        the values of the interpolation functions F_(k) of the mobility        reduction factor between the value M_(mod) ^(opt) and 1, which        each depend on a parameter V_(k) relative to at least one        characteristic of the foam, and which involve a certain number        of calibration constants to be calibrated as explained        hereinbelow.

In order to provide a model of displacement of the foam to the simulatorthat is representative of the reality, the method according to theinvention aims to determine, reliably from representative displacementmeasurements, the following modelling data:

-   -   the optimal mobility reduction factor of the gas M_(mod) ^(opt)        as defined according to the equation (3);    -   the calibration constants of each of the functions F_(k)        considered in the definition of the model of displacement of the        foam according to the equations (1) and (2).

According to one implementation of the invention, the parameter V_(k)can notably be the foaming agent concentration C_(s) ^(w), the watersaturation S_(w), the oil saturation S_(o), or even the gas flow rateu_(g.)

According to one implementation of the invention, the gas mobilityreduction functional, denoted FM, comprises four interpolation functionsF_(k)(V_(k)) and each of these functions comprises two constants to becalibrated from experimental data. According to an implementation of theinvention in which the gas mobility reduction functional comprises fourinterpolation functions F_(k)(V_(k)), the following are defined:

-   -   the interpolation function F₁ relative to the parameter V₁=C_(s)        ^(w) (foaming agent concentration C_(s) ^(w)) by a formula of        the type:

$\begin{matrix}{F_{1} = \left( \frac{{Min}\left( {C_{s}^{w},C_{s}^{w\text{-}{ref}}} \right)}{C_{s}^{w\text{-}{ref}}} \right)^{e_{s}}} & (4)\end{matrix}$

-   -   and for which the constants to be calibrated are the exponent        e_(s), and the constant C_(s) ^(w-ref) which corresponds to the        foaming agent concentration in reference optimal conditions;    -   the interpolation function F₂ relative to the parameter V₂=s_(w)        (water saturation), by a formula of the type:

$\begin{matrix}{F_{2} = \left\lbrack {0.5 + \frac{\arctan \left\lbrack {f_{w}\left( {S_{w} - S_{w}^{*}} \right)} \right\rbrack}{\pi}} \right\rbrack} & (5)\end{matrix}$

-   -   and for which the constants to be determined are the constant        f_(w) which governs the transition (according to the water        saturation) between the foaming and non-foaming states and the        constant S_(w)* which represents the transition water saturation        between stable and unstable foaming states;    -   the interpolation function F₃ relative to the parameter V₃=S_(o)        (oil saturation) by a formula of the type:

$\begin{matrix}{F_{3} = \left( \frac{{Max}\left\lbrack {0;{S_{o}^{*} - S_{o}}} \right\rbrack}{S_{o}^{*}} \right)^{e_{0}}} & (6)\end{matrix}$

-   -   in which S_(o)* is the oil saturation beyond which the foam        loses all its faculties to reduce the mobility of the gas, and        the exponent e_(o) is a constant to be determined;    -   the interpolation function F₄ relative to the parameter V₄=u_(g)        (gas flow rate) by a formula of the type:

$\begin{matrix}{F_{4} = {{\left( \frac{N_{c}^{*}}{{Max}\left( {N_{c},N_{c}^{*}} \right)} \right)^{e_{c}}\mspace{14mu} {with}\mspace{14mu} N_{c}} = \frac{\mu_{g}u_{g}}{{\varphi\sigma}_{vg}\left( C_{s}^{w} \right)}}} & (7)\end{matrix}$

-   -   in which N_(c)* is the reference value of the capillary number        N_(c) calculated for the reference optimal flow rate. The        variables involved in the calculation of N_(c) are the velocity        of the gas u_(g), the porosity φ of the formation considered,        the water-gas interfacial tension σ_(gw) (which is dependent on        the foaming agent concentration C_(s) ^(w)), and the viscosity        of the gas μ_(g). The exponent e_(c) is also a constant to be        calibrated.

Generally, it can be shown that any interpolation function F_(k) of theparameter V_(k) can be written in the form:

$\begin{matrix}{{F_{k}\left( V_{k} \right)} = {\frac{\frac{1}{F\; M} - 1}{\frac{1}{F\; M_{opt}} - 1} = \frac{{M_{mod}\left( V_{k} \right)} - 1}{M_{mod}^{opt} - 1}}} & (8)\end{matrix}$

in which M_(mod)(V_(k)) is the reduction of mobility for a value V_(k)of the parameter k affecting the foam (and for optimal values of theother parameters V_(j), j being different from k) and in which M_(mod)^(opt)=M_(mod)(V_(k) ^(opt)) is the reduction of mobility obtained forthe optimal value V_(k) ^(opt) of the parameter V_(k). The methodaccording to the invention thus consists, for each parameter V_(k)affecting the foam, in determining the factors M_(mod)(V_(k)) forvarious values of this parameter, and M_(mod) ^(opt), then indetermining, from these factors, the constants of the interpolationfunction F_(k) considered.

According to an implementation of the invention in which the functionalFM defined in the equation (2) involves the interpolation functions F₁,F₂, F₃ and F₄ defined in the equations (4) to (7), the determination ofthe model of displacement of the foam entails calibrating the 8constants: C_(s) ^(w-ref), e_(s), f_(w), S_(w)*, S_(o)*, e_(o), N_(c)^(ref), e_(c).

According to the invention, the determination of the constants of theinterpolation functions F_(k) involved in the equation (2) is performedvia a calibration, interpolation function by interpolation function (andnot globally, for all the functions), from experimental measurementsrelative to each of the interpolation functions, performed in theoptimal conditions established for the other interpolation functions.

The method according to the invention comprises at least the followingsteps, the step 1 being able to be repeated for each of theinterpolation functions of the model of displacement of the foam, andthe step 2 being able to be optional:

1. Laboratory measurements relative to an interpolation function

-   -   1.1. Definition of values of the parameter relative to the        interpolation function    -   1.2. Injections with/without foam and measurements of headlosses    -   1.3. Determination of an optimal parameter value

2. Laboratory measurements according to optimal conditions

3. Determination of the foam displacement model

-   -   3.1. Determination of the optimal mobility reduction factor    -   3.2. Calibration of the constants of the interpolation functions

4. Operation of the hydrocarbons of the formation

The various steps of the method according to the invention are detailedhereinbelow.

1. Laboratory Measurements Relative to an Interpolation Function

During this step, laboratory experiments are performed relative to agiven interpolation function F_(k) of the model of displacement of thefoam defined according to the equations (1) and (2). According to oneimplementation of the invention, this step is repeated for each of theinterpolation functions involved in the model of displacement of thefoam defined according to the equations (1) and (2). It should be notedthat the model of displacement of the foam can however comprise only asingle interpolation function (case for which k=1). Optimal values areadopted for the other parameters affecting the foam in such a way thatthe other interpolation functions F_(j), j different from k, have thevalue 1 or are invariant in these experiments relative to theinterpolation function F_(k).

During this step, applied to each interpolation function independentlyof one another, a plurality of values of the parameter is definedrelative to the interpolation function considered, then an injectioninto said sample of the gas in non-foaming form and of the gas in foamform is performed according to the values of the parameter relative tothe interpolation function considered, and a headloss with foam and aheadloss without foam are measured respectively for each of the valuesof the parameter relative to this function. This step is detailedhereinbelow for a given interpolation function F_(k).

1.1. Definition of Parameter Values Relative to the InterpolationFunction

During this substep, the aim is to define a plurality of values V_(k,i)(with i lying between 1 and I, and I>1) of the characteristic parameterV_(k) of the interpolation function F_(k) considered.

According to one implementation of the invention, the aim is to define arange of values of this parameter and a sampling step for this range.

According to one implementation of the invention, the plurality ofvalues of the parameter V_(k) relative to the interpolation functionF_(k) considered are defined from the possible or realistic values ofthe parameter considered (for example, a mass concentration of foamingagent less than 1% in all cases) and so as to sample in an ad hoc mannerthe curve representative of the interpolation function considered (aninterpolation function that has a linear behavior does not need a highnumber of measurements, unlike other types of function). The expert infoam injection-assisted recovery has a perfect knowledge of how todefine a plurality of ad hoc values of the parameters of each of theinterpolation functions F_(k). According to an implementation of theinvention in which the interpolation function considered relates to thefluid flow rate (parameter V₄ of the function F₄ of the equation (7)),an injection flow rate on coring of between 10 and 40 cm³/h is forexample chosen, with a step of 10 cm³/h.

1.2. Injections with/without Foam and Headloss Measurements

During this substep, at least two series of experiments are carried outon at least one sample of the underground formation for theinterpolation function F_(k) considered:

-   -   injection of gas in non-foaming form (more specifically, a        co-injection of water and of gas in non-foaming form) into the        sample considered for each of the values V_(k,i) of the        parameter V_(k) relative to the function F_(k) considered. The        flow rates of gas and of water adopted for each of these        co-injections are the same as the flow rates of gas and of water        injected in foam form in the tests which follow these        co-injections. For example, in the case of the interpolation        function F₄ of the equation (7), the flow rate only is made to        vary in the sample considered, the parameters of the other        interpolation functions F₁, F₂, F₃ (for example, the foaming        agent concentration, the quality of the foam and the oil        saturation) being fixed. During each of the experiments of this        first series, a headloss (that is to say a pressure difference)        is measured, that is denoted ΔP_(k,i) ^(NOFO), for each value        V_(k,i);    -   injection of foam: the same experiment is repeated, for the same        values of the parameter considered (for example the flow rate        for the interpolation function F₄ according to the equation        (7)), but by this time injecting the water and the gas in foam        form. During each of the experiments of this second series, a        headloss (that is to say a pressure difference) is measured,        that is denoted ΔP_(k,i) ^(FO), pr for each value V_(k,i);

According to one implementation of the invention, the injections of gasin non-foaming form and in foam form are performed on samples of theformation initially saturated with a liquid phase (such as water and/oroil), the latter being able to be mobile or residual depending on thehistory of the coring and the measurement objectives (checking mobilityof the gas in secondary or tertiary injection, after injection ofwater). The displacements studied are then draining processes in whichthe saturation of the gas phase increases in all cases.

According to a variant implementation of the invention, it is possibleto measure, in addition to the headlosses, the productions of liquidphase (water and/or oil) and of gas, and, possibly, the gas saturationprofiles during the transitional period of the displacement and in thesteady state. These optional measurements make it possible to validatethe model once the interpolation functions F_(k) are calibrated.

1.3. Determination of an Optimal Parameter Value

During this substep, the aim is to determine the value V_(k) ^(opt),that will hereinafter be called optimal value, maximizing the ratiobetween the headlosses without foam ΔP_(k,i) ^(NOFO) and the headlosseswith foam ΔP_(k,i) ^(FO) relative to the interpolation function F_(k)considered and measured during the preceding substep. Thus, if M_(lab)^(k,i) is used to denote the ratio of the headlosses measured in thepresence and in the absence of foam for the value V_(k,i) of theparameter V_(k), i.e.

${M_{lab}^{k,i} = {\frac{\Delta \; P_{k,t}^{FO}}{\Delta \; P_{k,i}^{NOFO}} = \frac{k_{rg}\left( S_{g{({k,i})}}^{NOFO} \right)}{k_{rg}^{FO}\left( S_{g{({k,i})}}^{FO} \right)}}},$

it is then possible to define the optimal value V_(k,iopt) as the valueV_(k,i) which maximizes M_(lab) ^(k,i) whose value is then denoted asfollows:

$\begin{matrix}{M_{lab}^{k,{iopt}} = {M_{lab}^{kopt} = {\underset{i}{Max}M_{lab}^{k,i}}}} & (9)\end{matrix}$

According to a preferred implementation of the invention, the step 1 asdescribed hereinabove is repeated for each of the parameters V_(k)relative to each of the interpolation functions F₁, taken intoconsideration for the implementation of the method according to theinvention. Thus, at the end of such a repetition, an optimal value V_(k)^(opt) is obtained for each parameter V_(k).

Subsequently, “optimal conditions” is the term used to denote the set ofthe values V_(k) ^(opt) determined on completion of the step 1, thelatter if necessary being repeated for each of the interpolationfunctions taken into consideration for the implementation of the methodaccording to the invention.

2. Laboratory Measurements According to Optimal Conditions

During this step, the aim is to perform two types of experiments on atleast one sample of the underground formation, by injecting gas innon-foaming form, and gas in foam form, similarly to the substep 1.2,but this time in the optimal conditions determined on completion of thesubstep 1.3, this substep being repeated if necessary for each of theinterpolation functions taken into consideration for the definition ofthe model of displacement of the foam according to the equations (1) and(2)). In other words, the following measures are carried out:

-   -   injection of gas in non-foaming form (more specifically, a        co-injection of water and of gas in non-foaming form) into the        sample considered, this injection being performed in the optimal        conditions (defined by the set of the optimal values V_(k)        ^(opt) determined for each parameter V_(k)) determined on        completion of the step 1. During this first experiment, a        headloss (that is to say a pressure difference) is measured,        that is then denoted ΔP_(opt) ^(NOFO);    -   injection of foam (that is to say an injection of gas and of        water, with an addition of a foaming agent into one of the water        or gas phases) into the sample considered, this injection being        performed in the optimal conditions (defined by the set of the        optimal values V_(k) ^(opt) determined for each parameter V_(k))        determined on completion of the step 1. During this second        experiment, a headloss (that is to say a pressure difference) is        measured, that is then denoted ΔP_(opt) ^(FO).

Subsequently, M_(lab) ^(opt) will be used to denote the optimal mobilityreduction factor relative to the laboratory measurements, defined by aformula of the type:

$\begin{matrix}{M_{lab}^{opt} = {\frac{\Delta \; P_{opt}^{FO}}{\Delta \; P_{opt}^{NOFO}} = \frac{k_{rg}\left( S_{g,{opt}}^{NOFO} \right)}{k_{rg}^{FO}\left( S_{g,{opt}}^{FO} \right)}}} & (10)\end{matrix}$

This step is not necessary in practice if the precaution to perform theexperiments of the step 1 relative to each of the parameters V_(k) byadopting optimal values V_(j,j≠k) ^(opt) of the other parameters V_(j)affecting the foam has indeed been taken. This step nevertheless makesit possible to refine the value of M_(lab) ^(opt), if the assumedoptimal conditions of the parameters V_(j,j≠k) were not perfectlysatisfied.

3. Determination of the Foam Displacement Model

3.1. Determination of the Optimal Mobility Reduction Factor

During this substep, the aim is, from the headloss measurementsperformed in the optimal conditions, from measurements of conventionalrelative permeabilities to the gas in non-foaming form and frommeasurements of conventional relative permeabilities to the aqueousphase, to determine an optimal mobility reduction factor, that is to saythe factor of reduction of the relative permeabilities to the gas when,present at a given saturation within the porous medium, it circulates infoam form or in continuous phase form (in the presence of water).

According to one implementation of the invention, the optimal mobilityreduction factor is determined according to at least the followingsteps:

-   -   from the conventional relative permeabilities to the gas k_(rg)        and to the aqueous phase k_(rw), the gas saturation is        calculated in steady state conditions of flow of non-foaming gas        and water S_(g) ^(NOFO) according to a formula of the type:

$\begin{matrix}{S_{g}^{NOFO} = {\left( \frac{k_{rg}}{k_{rw}} \right)^{- 1}\left( {\frac{f_{g}}{1 - f_{g}}\frac{\mu_{g}}{\mu_{w}}} \right)}} & (11)\end{matrix}$

-   -   in which f_(g) is the fractional gas flow rate (ratio of the gas        flow rate to the total flow rate), μ_(g) and μ_(w) are,        respectively, the viscosity of the gas and of the water;    -   from the ratio of headlosses measured in the optimal conditions        as defined on completion of the step 1 (the step 1 being able to        be repeated if necessary for each of the interpolation functions        F_(k) considered), from the gas saturation in steady state        conditions of flow of non-foaming gas and water S_(g) ^(NOFO),        the gas saturation in the presence of foam S_(g) ^(FO) is        calculated according to a formula of the type:

$\begin{matrix}{S_{g,{opt}}^{FO} = {1 - {\left( k_{rw} \right)^{- 1}\left\{ \frac{k_{rw}\left( {S_{w}^{NOFO} = {1 - S_{g}^{NOFO}}} \right)}{M_{lab}^{opt}} \right\}}}} & (12)\end{matrix}$

-   -   This relationship devolves from the known assumption of        invariance of the functions of relative permeability to water        flowing in the form of foam films or in conventional continuous        form.    -   from the gas saturation in steady state conditions of flow of        non-foaming gas and water S_(g) ^(NOFO) from the gas saturation        in the presence of foam S_(g) ^(FO), in the optimal conditions,        from the factor M_(lab) ^(opt) determined in the optimal        conditions (see step 2), the mobility reduction factor M_(mod)        ^(opt) is determined according to a formula of the type:

$\begin{matrix}{M_{mod}^{opt} = {M_{lab}^{opt}\frac{k_{rg}\left( S_{g,{opt}}^{FO} \right)}{k_{rg}\left( S_{g,{opt}}^{NOFO} \right)}}} & (13)\end{matrix}$

3.2. Calibration of the Constants of the Interpolation Functions

During this substep, the constants of each of the interpolationfunctions F_(k) considered are calibrated, from the optimal mobilityreduction factor M_(mod) ^(opt), from the headloss measurements relativeto the interpolation function considered, from the measurements ofconventional relative permeabilities to the gas in non-foaming form andfrom the measurements of conventional relative permeabilities to theaqueous phase.

According to one implementation of the invention, the proceduredescribed in the substep 3.1 is applied beforehand to the ratios of theheadlosses M_(lab) ^(k,i) measured in the presence and in the absence offoam for the different values V_(k,i) of the parameter V_(k). Thus, themobility reduction factors M_(mod) ^(k,i) relative to the values V_(k,i)of the parameter V_(k) are determined according to a formula of thetype:

$\begin{matrix}{M_{mod}^{k,i} = {M_{lab}^{k,i}\frac{k_{rg}\left( S_{g,{({k,i})}}^{FO} \right)}{k_{rg}\left( S_{g{({k,i})}}^{NOFO} \right)}}} & (14)\end{matrix}$

in which the gas saturation in the presence of foam S_(g(k,i)) ^(FO) forthe values V_(k,i) of the parameter V_(k) is obtained according to aformula of the type:

$\begin{matrix}{S_{g{({k,i})}}^{FO} = {1 - {\left( k_{rw} \right)^{- 1}\left\{ \frac{k_{rw}\left( {S_{w{({k,i})}}^{NOFO} = {1 - S_{g,{({k,i})}}^{NOFO}}} \right)}{M_{lab}^{k,i}} \right\}}}} & (15)\end{matrix}$

Advantageously, this operation is repeated for each of the interpolationfunctions F_(k). Then, the constants of each of the interpolationfunctions F_(k) considered are calibrated, from the optimal mobilityreduction factor M_(mod) ^(opt) and from the values of the mobilityreduction factors M_(mod) ^(k,i) relative to each interpolation functiondetermined as described above. In the case of the function F₄ forexample, a value of the exponent e_(c) is determined which most closelyadjusts the values of M_(mod) ^(4,i) corresponding to the values V_(4,i)of the parameter studied (flow rate in this example), which isformulated as follows:

${F_{4}\left( V_{4,i} \right)} = {\left( \frac{N_{c}^{*}}{{Max}\left( {N_{c,i},N_{c}^{*}} \right)} \right)^{e_{c}} = \frac{M_{mod}^{4,i} - 1}{M_{mod}^{opt} - 1}}$

According to one implementation of the invention, this calibration,interpolation function by interpolation function, can be performed by aleast squares method, such as for example an inverse method based on theiterative minimization of a functional. The expert has a perfectknowledge of such methods. Advantageously, the implementation of a leastsquares method and in particular the iterative minimization of afunctional, is performed by means of a computer.

According to another implementation of the invention, such a calibrationis carried out, interpolation function by interpolation function,graphically. The expert has a perfect knowledge of such methods forcalibrating constants of a function from a series of values of saidfunction.

Thus, on completion of this step, there is a model of displacement ofthe foam that is calibrated and suitable for use by an ad hoc flowsimulator.

4. Operation of the Hydrocarbons

During this step, the aim is to define at least an optimal scheme foroperating the fluid contained in the formation, that is to say anoperating scheme that allows for an optimal operation of a fluidconsidered according to technical and economic criteria predefined bythe expert. It can be a scenario offering a high rate of recovery of thefluid, over a long period of operation, and requiring a limited numberof wells. Then, according to the invention, the fluid of the formationstudied is operated according to this optimal operation scheme.

According to the invention, the determination of said operational schemeis performed using a flow simulation that makes use of the foamdisplacement model established during the preceding steps. One exampleof flow simulator that makes it possible to take account of a foamdisplacement model is the PumaFlow software (IFP Energies nouvelles,France).

According to the invention, at any instant t of the simulation, the flowsimulator solves all the flow equations specific to each mesh anddelivers solution values of the unknowns (saturations, pressures,concentrations, temperature, etc.) predicted at that instant t. Theknowledge of the quantities of oil produced and of the state of thedeposit (distribution of the pressures, saturations, etc.) at theinstant considered devolves from this resolution. According to oneimplementation of the invention, different schemes for operating thefluid of the formation studied are defined and the flow simulatorincorporating the foam displacement model determined on completion ofthe step 3 is used to estimate the quantity of hydrocarbons producedaccording to each of the different operating schemes.

An operating scheme relative to a foam injection-assisted recovery cannotably be defined by a type of gas injected into the formation studiedand/or by the type of foaming agent added to this gas, by the quantityof foaming agent, etc. An operating scheme is also defined by a number,a geometry and a layout (position and spacing) of the injecting andproducing wells in order to best take account of the impact of thefractures on the progression of the fluids in the reservoir. In order todefine an optimal operating scheme, various tests of differentproduction scenarios can be performed using a flow simulator. Theoperating scheme that offers the best fluid recovery rate for the lowestcost will for example be preferred. By selecting various scenarios,characterized for example by various respective layouts of the injectingand producing wells, and by simulating the fluid production for each ofthem, it is possible to select the scenario that makes it possible tooptimize the production of the formation considered according to thetechnical and economic criteria predefined by the expert. The operatingscheme offering the best fluid recovery rate for the lowest cost willfor example be considered as the optimal operating scheme.

The experts then operate the fluid of the formation considered accordingto the scenario that makes it possible to optimize the production fromthe deposit, notably by drilling injecting and producing wells definedby said optimal operating scheme, and to produce the fluid according tothe recovery method defined by said optimal operating scheme.

Production Example

The features and advantages of the method according to the inventionwill become more clearly apparent on reading about the followingexemplary application.

More specifically, the present invention has been applied to anunderground formation in which the reservoir rock consists of sandstone,of the Berea sandstone type. An assisted recovery of the hydrocarbonscontained in the reservoir based on an injection of foaming CO2 istrialed.

For this example, a functional FM of the foam displacement model is usedaccording to the equation (2) defined by the four interpolationfunctions according to the equations (4) to (7). As prescribed in themethod according to the invention, the calibration of the constants ofthe interpolation functions is carried out interpolation function byinterpolation function. Only the calibration of the interpolationfunction F₄ (see equation (7)) is detailed hereinbelow, but the sameprinciple can be applied to the other interpolation functions.

According to the step 1.2 described above, a series of co-injections ofgas and of water and of injections of foam were carried out in thelaboratory, on a sample of the reservoir rock originating from theformation studied. The characteristics of this sample and themeasurement conditions are presented in Table 1. A non-dense gaseousmixture consisting of 62% CO2 and 38% methane at a temperature of 100°C. and a pressure of 100 bar was injected. These displacements wereperformed with fixed fractional gas flow rate (equal to 0.8) and fordifferent successive total flow rates (10, 20, 30 and 40 cm³/h). The oilis absent for this series of tests and the headlosses in steady stateconditions of flow of water and of gas on the one hand, of foam on theother hand, were measured in the same conditions.

The conventional relative permeabilities required to resolve theequations (11) and (12) are analytical functions defined as powerfunctions (called Corey functions) with exponents equal to approximately2.5 for the gas and 3.9 for the water with an irreducible drainage watersaturation equal to 0.15, and limit points equal to 0.2 for the gas and1 for the water, i.e.:

${k_{rg}\left( S_{g}^{NOFO} \right)} = {0.2\left( \frac{S_{g}^{NOFO}}{0.85} \right)^{2.5}}$${k_{rw}\left( S_{w}^{NOFO} \right)} = \left( \frac{S_{w}^{NOFO} - 0.15}{0.85} \right)^{3.9}$

These curves of relative permeabilities were estimated beforehand fromliterature data and checked afterwards by comparison of the headlossvalues calculated and measured during the co-injections of gas and ofwater. In this way the model of relative permeability to the foam doesindeed return the gas mobility reductions, i.e. the ratios of relativepermeability in the absence and in the presence of foam, but withoutnecessarily well reproducing the real diphasic behavior (transientstates in particular).

Table 1 presents the headlosses (pressure gradient) with and withoutfoam for four values of the parameter V₄=u_(g) of the equation (7). Fromthese values, the value of

$M_{lab}^{kopt} = {\underset{i}{Max}\; M_{lab}^{k,i}}$

is deduced therefrom. This value (M_(lab) ^(4opt) in this example),equal to 83, was obtained for a flow rate V₄ ^(opt) equal to 20 cm³/h(see substep 1.3). These laboratory experiments were repeated for theother parameters of the other interpolation functions F₁, F₂, and F₃.The optimal conditions are then determined for all of the interpolationfunctions.

According to the step 2, measurements are performed with and withoutfoam in the duly determined optimal conditions. The optimal mobilityreduction factor M_(mod) ^(opt) is then determined, in accordance withthe step 3.1, as are the values of the mobility reduction factorsM_(mod) ^(k,i) relative to the sampled values V_(k,i) of the parameterV_(k), in accordance with the step 3.2. The calibration of each of theinterpolation functions is then carried out. In particular, the constante_(c) of the function F₄ was calibrated and a value close to 0.6 wasdetermined.

FIG. 1 shows by a solid line the trend of the mobility reduction factorR as a function of the parameter V₄ of the function F₄ (flow rate Q)deduced from the method according to the invention. The comparison withthe mobility reduction factor deriving from the flow simulation (dottedline curve) shows a good consistency with the foam displacement model ofaccording to the invention.

Thus, the method according to the invention allows for a reliabledetermination of the foam displacement model from experimental dataproduced and processed according to a sequential and systematicapproach, parameter by parameter, and not by the overall adjustment of aset of measurements varying one or more parameters simultaneously.Moreover, given the parametric complexity of the behavior of the foams,the experiments according to the method according to the invention arecarried out in conditions as close as possible to the reservoirconditions.

TABLE 1 Berea sandstone Total flow rate [cm3/h] 10 20 30 40 P = 100 barΔP_(k,i) ^(FO) 5.2 10.3 11.9 13.6 T = 100° C. ΔP_(k,i) ^(NOFO) 0.070.124 0.18 0.235 L = 15 cm A = 12.56 cm2 φ = 0.19 Kw = 120 mD μw = 0.28cp μg = 0.02 cp ρg = 0.125 g/cm3$M_{lab}^{k,i} = \frac{\Delta \; P_{k,i}^{NOFO}}{\Delta \; P_{k,i}^{FO}}$74 83 66 58

1. Method for operating an underground formation comprisinghydrocarbons, by means of an injection of an aqueous solution comprisinga gas in foam form and a flow simulator relying on a displacement modelof said gas in foam form, said displacement model being a function of anoptimal mobility reduction factor of said gas and of at least oneinterpolation function of said optimal mobility reduction factor, saidinterpolation function being a function of at least one parameterrelating to at least one characteristic of the foam and of at least oneconstant, characterized in that, from at least one sample of saidformation, measurements of conventional relative permeabilities to saidgas in non-foaming form and measurements of conventional relativepermeabilities to said aqueous phase: A. said displacement model of saidsimulator is determined according to at least the following steps: i. aplurality of values of said parameter is defined relative to at leastone of said interpolation functions, an injection is performed into saidsample of said gas in non-foaming form and of said gas in foam formaccording to said values of said parameter relative to said function,and a headloss with foam and a headloss without foam are measuredrespectively for each of said values of said parameter relative to saidfunction; ii. from said measurements of headloss relative to saidinterpolation function, an optimal value of said parameter relative tosaid function is determined, said optimal value making it possible tomaximize a ratio between said headlosses without foam and saidheadlosses with foam measured for said function; iii. from saidmeasurements of headloss performed with said optimal value determinedfor at least said interpolation function, from said measurements ofconventional relative permeabilities to said gas in non-foaming form andfrom said measurements of conventional relative permeabilities to saidaqueous phase, said optimal mobility reduction factor is determined; iv.for at least said interpolation function, from said optimal mobilityreduction factor, from said measurements of headloss relative to saidinterpolation function, from said measurements of conventional relativepermeabilities to said gas in non-foaming form and from saidmeasurements of conventional relative permeabilities to said aqueousphase, said constants of said interpolation function are calibrated;B—from said displacement model and from said flow simulator, an optimaloperation scheme for said deposit is determined and said hydrocarbonsare exploited.
 2. Method according to claim 1, in which saiddisplacement model of the foam is expressed in the form:k _(rg) ^(FO)(S _(g))=FM k _(rg)(S _(g)) in which k_(rg) ^(FO)(S_(g)) isthe relative permeability to said gas in foam form for a given gassaturation value S_(g), k_(rg)(S_(g)) is the relative permeability tosaid non-foaming gas for said gas saturation value S_(g), and FM is afunctional expressed in the form:${F\; M} = \frac{1}{1\left( {M^{opt} - 1} \right)*{\prod\limits_{k}\; F_{k\,}}}$in which M^(opt) is said optimal mobility reduction factor of said gasand F_(k) is one of said interpolation functions, with k≧1.
 3. Methodaccording to claim 1, in which there are four of said interpolationfunctions and said parameters of said functions are a foaming agentconcentration, a water saturation, an oil saturation and a gas flowrate.
 4. Method according to claim 1, in which said interpolationfunction F_(k) of a parameter V_(k) is written in the form:${F_{k}\left( V_{k} \right)} = \frac{M_{k}^{opt} - 1}{M^{opt} - 1}$ inwhich M^(opt) is said optimal mobility reduction factor and M_(k) ^(opt)is an optimal mobility reduction factor for said parameter V_(k). 5.Method according to claim 1, in which, prior to the step iii), optimalconditions are defined as corresponding to said optimal valuesdetermined for each of said interpolation functions, said gas innon-foaming form and said gas in foam form are injected into said sampleaccording to said optimal conditions, and a headloss with foam and aheadloss without foam are respectively measured.
 6. Method according toclaim 1, in which said constants of at least one of said interpolationfunctions are calibrated by a least squares method, such as an inversemethod based on the iterative minimization of a functional.